The perturbation, and the overall strategy of proof, closely follows the analysis in [1]} for the case of flat three-manifolds, with a significant complication: their argument strongly relies of the fact that on a flat three-manifold harmonic spinors and 1-forms are parallel, hence are determined by their value at a given point. This is not true for the manifolds we consider; on the other hand, we will see that the assumption \(\Sigma /\varphi =\mathbb {P}^1\)  allows to reach the same conclusions via the study of the pulled-back equations on \(\Sigma \times S^1\)  and the action of \(\varphi \) .
